Since \(\lim _{N \rightarrow \infty }\)  Beta\((\alpha =1,\beta =N) = \exp (\lambda = N),\)  we approximate \(t_{mi}\sim \exp (N)\) , which is a good approximation at high \(N\)  [1]}.
Even with this approximation, \(u_m\)  is identically Gamma distributed across users but not independent.
Thus, with the independence assumption, \(u_m\)  is i.i.d. Gamma distributed with shape parameter \(r-1\)  and rate parameter \(N\) , i.e., \( u_m \mathrel {\overset{\text{i.i.d.}}{\scalebox {1.5}[1]{\sim }}}\)  Gamma\((r-1,N)\) .
Thus, we obtain the Gamma approximation:
\(\theta _r & \approx 1 - \Gamma _{\text{inc}}(r-1,N t_0)/\Gamma (r-1). \) 
